Exceptional Jordan Strings/Membranes

and Octonionic Gravity/p-branes

Carlos CastroCenter for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta, Georgia. 30314, perelmanc@hotmail.com

March 2011

Abstract

Nonassociative Octonionic Ternary Gauge Field Theories are revisitedpaving the path to an analysis of the many physical applications of Ex-ceptional Jordan Strings/Membranes and Octonionic Gravity. The oldoctonionic gravity constructions based on the split octonion algebra Os

(which strictly speaking is not a division algebra) is extended to the fullfledged octonion division algebra O. A real-valued analog of the Einstein-Hilbert Lagrangian L = R involving sums of all the possible contractionsof the Ricci tensors plus their octonionic-complex conjugates is presented.A discussion follows of how to extract the Standard Model group (thegauge fields) from the internal part of the octonionic gravitational con-nection. The role of Exceptional Jordan algebras, their automorphismand reduced structure groups which play the roles of the rotation andLorentz groups is also re-examined. Finally, we construct (to our knowl-edge) generalized novel octonionic string and p-brane actions and raisethe possibility that our generalized 3-brane action (based on a quarticproduct) in octonionic flat backgrounds of 7, 8 octonionic dimensions maydisplay an underlying E7, E8 symmetry, respectively. We conclude withsome final remarks pertaining to the developments related to Jordan ex-ceptional algebras, octonions, black-holes in string theory and quantuminformation theory.

Keywords: Octonions, ternary algebras, Lie 3-algebras, membranes, nonasso-ciative gauge theories, nonassociative geometry, Exceptional Jordan Algebras,Exceptional Groups, Grand Unification.

1 Introduction

Exceptional, Jordan, Division, Clifford, noncommutative and nonassociative al-gebras are deeply related and are essential tools in many aspects in Physics, see

1

[1], [2], [3], [4], [7], [8], [9], for references, among many others.A thorough discussion of the relevance of ternary and nonassociative struc-

tures in Physics has been provided in [5], [10], [11]. The earliest example ofnonassociative structures in Physics can be found in Einstein’s special theoryof relativity. Only colinear velocities are commutative and associative, butin general, the addition of non-colinear velocities is non-associative and non-commutative.

Recently, tremendous activity has been launched by the seminal works ofBagger, Lambert and Gustavsson (BLG) [12], [13] who proposed a Chern-Simons type Lagrangian describing the world-volume theory of multiple M2-branes. The original BLG theory requires the algebraic structures of generalizedLie 3-algebras and also of nonassociative algebras. Later developments by [14]provided a 3D Chern-Simons matter theory with N = 6 supersymmetry andwith gauge groups U(N)× U(N), SU(N)× SU(N). The original constructionof [14] did not require generalized Lie 3-algebras, but it was later realized that itcould be understood as a special class of models based on Hermitian 3-algebras[15], [16]. For more recent developments we refer to [17] and references therein.

In this work we explore further physical applications of Exceptional JordanStrings/Membranes and Octonionic Gravity [24], [27], [18], [19]. The outline ofthis work is organized as follows. In section 2 we present a review of OctonionicTernary Gauge Field Theories [28] and add new material pertaining octonionic-valued SU(N) Yang-Mills and 3-Lie-algebra gauge field theories.

In section 3 we shall generalize the octonionic gravity construction based onthe split octonion algebra Os (which strictly speaking is not a division algebra)studied by [18], [19] to the full fledged octonion division algebra O. A real-valued analog of the Einstein-Hilbert Lagrangian L = R involving sums ofall the possible contractions of the Ricci tensors plus their octonionic-complexconjugates is presented. Section 3 ends with a discussion of how to extractthe Standard Model group (the gauge fields) from the internal part of theoctonionic gravitational connection. The role of Exceptional Jordan algebras,their automorphism and reduced structure groups which play the roles of therotation and Lorentz groups is also examined.

Finally, in section 4, we briefly discuss Exceptional Jordan Strings/Membranesand provide a series of generalized octonionic string and p-brane actions (thatare novel to our knowledge) and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of7, 8 octonionic dimensions may display an underlying E7, E8 symmetry, respec-tively. We conclude with some final remarks pertaining to the developmentsrelated to Jordan exceptional algebras, octonions, black-holes in string theoryand quantum information theory.

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2 Octonionic Ternary Gauge Field Theories

Recently [28] , a novel (to our knowledge) nonassociative and noncommutativeoctonionic ternary gauge field theory was explicitly constructed that it is basedon a ternary-bracket structure involving the octonion algebra. The ternarybracket obeying the fundamental identity (generalized Jacobi identity) was de-veloped earlier by Yamazaki [29]. The field strength Fµν = ∂[νAµ] − [Aµ, Aν ,g]is defined in terms of the 3-bracket [Aµ, Aν ,g] involving an octonionic-valuedfield Aµ = (Aµ)aea, and an octonionic-valued coupling g = gaea. In this sectionwe shall review briefly the Octonionic Ternary Gauge Field Theory description[28] and add some new material.

Given an octonion X it can be expanded in a basis (eo, em) as

X = xo eo + xm em, m, n, p = 1, 2, 3, .....7. (2.1)

where eo is the identity element. The Noncommutative and Nonassociativealgebra of octonions is determined from the relations

e2o = eo, eoei = eieo = ei, eiej = −δijeo + cijkek, i, j, k = 1, 2, 3, ....7. (2.2)

where the fully antisymmetric structure constants cijk are taken to be 1 forthe combinations (124), (235), (346), (457), (561), (672), (713) [30]. The octonionconjugate is defined by eo = eo, em = −em

X = xo eo − xm em. (2.3)

and the norm is

N(X) = | < X X > | 12 = | Real (X X) | 12 = | (xo xo + xk xk) | 12 . (2.4)

The inverse

X−1 =X

< X X >, X−1X = XX−1 = 1. (2.5)

The non-vanishing associator is defined by

(X,Y,Z) = (XY)Z−X(YZ) (2.6)

In particular, the associator

(ei, ej , ek) = (eiej)ek − ei(ejek) = 2 dijkl el

dijkl =13!

εijklmnp cmnp, i, j, k.... = 1, 2, 3, .....7 (2.7)

Yamazaki [29] define the three-bracket as

3

[ u, v, x ] ≡ Du,v x =12

( u(vx)− v(ux) + (xv)u − (xu)v + u(xv) − (ux)v ) .

(2.8)For the octonionic algebra, after a straightforward calculation when the indicesspan the imaginary elements a, b, c, d = 1, 2, 3, ......, 7, one has that

[ ea, eb, ec ] = fabcd ed = − [ dabcd − δac δbd + δbc δad ] ed (2.9a)

whereas[ ea, eb, e0 ] = [ ea, e0, eb ] = [ e0, ea, eb ] = 0 (2.9b)

The ternary bracket (2.8) obeys the fundamental identity

[ [x, u, v], y, z ] + [ x, [y, u, v], z ] + [ x, y, [z, u, v] ] = [ [x, y, z], u, v ](2.10)

A bilinear positive symmetric product < u, v >=< v, u > is required such thatthat the ternary bracket/derivation obeys what is called the metric compatibilitycondition

< [u, v, x], y > = − < [u, v, y], x > = − < x, [u, v, y] > ⇒

Du,v < x, y > = 0 (2.11)

The symmetric product remains invariant under derivations. There is also theadditional symmetry condition required by [29]

< [u, v, x], y > = < [x, y, u], y > (2.12)

Thus, the ternary product provided by Yamazaki (2.8) obeys the key fun-damental identity (2.10) and leads to the structure constants fabcd that arepairwise antisymmetric but are not totally antisymmetric in all of their indices: fabcd = −fbacd = −fabdc = fcdab; however : fabcd 6= fcabd; and fabcd 6= − fdbca.The associator ternary operation for octonions (x, y, z) = (xy)z − x(yz) doesnot obey the fundamental identity (2.10) as emphasized by [29]. For this reasonwe cannot use the associator to construct the 3-bracket.

We defined in [28] the ternary field strength in terms of the ternary bracketas

Fµν = ∂µBν − ∂νBµ + [ Bµ, Bν , g ] (2.13)

where g = gaea is an octonionic-valued ”coupling” function which is not inertunder octonionic gauge transformations. Only the scalar part of g remainsinvariant. It was shown in [28] after some algebra that under the local gaugetransformations

δ(Bmµ em) = Λab(x) [ea, eb, B

cµ ec] (2.14)

4

andδ(gm em) = Λab(x) [ea, eb, g

c ec] (2.15)

one can ensure that the ternary field strength Fµν defined in terms of the 3-brackets (2.13) transforms properly (homogeneously) under the ternary gaugetransformations if, and only if, the bivector gauge parameters Λab(x) obey the”self-duality” equations 1

2dabcmΛab(x) = Λcm(x). If this is so then Fµν trans-forms homogeneously under the infinitesimal ternary gauge transformations as

δ(Fmµν em) = Λab [ ea, eb, F c

µν ec ] = Λab F cµν f m

abc em ⇒ δFmµν = Λab F c

µν f mabc

(2.16)The result (2.16) is a direct consequence of the fundamental identity (2.10)because the 3-bracket (2.8) is defined as a derivation

[ [ea, eb, Bµ], Bν , g ] + [ Bµ, [ea, eb, Bν ], g ] + [ Bµ, Bν , [ea, eb, g] ] =

[ ea, eb, [Bµ, Bν , g] ] (2.17)

The parameter Λo(x) involved in the transformation δBoµ = ∂µΛo(x), corre-

sponding to the real (identity) element e0 of the octonion algebra, leads toδF 0

µν = 0 where the field strength component is Abelian-Maxwell-like F 0µν =

∂µB0ν − ∂νB0

µ.One can verify that the expression for U = exp (−αΛab[ea, eb]); U−1 = U =

exp (αΛab[ea, eb]), where one excludes the identity element e0 from the abovedefinition of U because it yields the trivial transformation U = 1, and α = 1

4 isa real numerical constant, yields the finite gauge transformations

F ′ = e−αΛab[ ea, eb ] (F c tc) eαΛab[ ta, tb ]. (2.18)

which agree with the ternary ones (2.17) when the real parameters Λab areinfinitesimals

δF = F ′ − F = Λab F c [ ea, eb, ec ] = − α Λab F c [ [ ea, eb ], ec] ⇒

Λab F c fabcm em = − α Λab F c (2cabd) (2cdcm) em ⇒ −4 α cabd cdcm = fabcm.(2.19)

Therefore, by choosing α = 14 one arrives at the condition among the structure

constants given by cabd cdcm = − fabcm and which is indeed obeyed for theoctonion algebra as shown in [30]; i.e. the Yamazaki 3-bracket (2.8) satisfies theidentity for octonions when a, b, c, m = 1, 2, 3, ....., 7

[ ea, eb, ec ] = fabcm em = − [ dabcm − δac δbm + δbc δam ] em =

− 14

[ [ ea, eb ], ec ] = − cabd cdcm em ⇒

cabd cdcm = dabcm − δac δbm + δbc δam (2.20)

5

dabcm are the associator structure constants given by the duals to the octonionstructure constants as shown in eq-(2.7). A series of identities involving thestructure constants of octonions can be found in [30]. Therefore, by choosingα = 1

4 , the equality in eq-(2.20) is indeed satisfied for the octonion algebraand such that for infinitesimal real valued parameters Λab eq-(2.18) yields tolowest order δF = F ′ − F = Λab[ea, eb, F ] recovering the homogeneous ternaryinfinitesimal gauge transformations for the field strengths as expected.

Given the octonionic valued field strength Fµν = F aµν ea , with real valued

components F 0µν , F i

µν ; i = 1, 2, 3, ....., 7, a gauge invariant action under ternaryinfinitesimal gauge transformations in D-dim is

S = − 14κ2

∫dDx < Fµν Fµν > (2.21)

κ is a numerical parameter introduced to make the action dimensionless andit can be set to unity for convenience. The < > operation is defined as< XY >= Real(XY ) =< Y X >= Real(Y X). Under infinitesimal ternarygauge transformations of the action one has

δ S = − 14

∫dDx < Fµν (δFµν) + (δFµν) Fµν > =

− 14

∫dDx < F c

µν ec Λab [ea, eb, Fµν n en] > +

− 14

∫dDx < Λab [ea, eb, F c

µν ec] Fµν n en > =

−14

∫dDx Λab F c

µν Fµν n ( < ec fabnk ek > + < fabck ek en > ) = 0.

(2.22)since

< ec fabnk ek > + < fabck ek en > = fabnk δck + fabck δkn = fabnc + fabcn =

− [ dabnc − δan δbc + δbn δac ] − [ dabcn − δac δbn + δbc δan ] = 0 (2.23)

because dabnc+dabcn = 0; dnabc+dcabn = 0, due to the total antisymmetry of theassociator structure constant dnabc under the exchange of any pair of indices. In-variance δS = 0, only occurs if, and only if, δF = Λab[ea, eb, F

cec] 6= Λab[F cec, ea, eb].The ordering inside the 3-bracket is crucial. One can check that if one setsδF = Λab[F cec, ea, eb], the variation δS leads to a term in the integral which isnot zero

fnabc + fcabn = − [ dnabc − δnb δac + δab δnc ]− [ dcabn − δcb δan + δab δcn ] 6= 0(2.24)

However, under δF = Λab[ea, eb, Fcec], the variation δS is indeed zero as shown.

This is a consequence of the fact that [ea, eb, ec] 6= [ec, ea, eb] when the 3-bracketis given by eq-(2.8).

6

To show that the action is invariant under finite ternary gauge transforma-tions requires to follow a few steps. Firstly, one defines

< x y > ≡ Real [ x y ] =12

( x y + y x ) ⇒ < x y > = < y x > (2.25)

Despite nonassociativity, the very special conditions

x(xu) = (xx)u; x(ux) = (xu)x; x(xu) = (xx)u; x(ux) = (xu)x (2.26)

are obeyed for octonions resulting from the Moufang identities. Despite that(xy)z 6= x(yz) one has that their real parts obey

Real [ (x y) z ] = Real [x (y z) ] (2.27)

Due to the nonassociativity of the algebra, in general one has that (UF )U−1 6=U(FU−1). However, if and only if U−1 = U ⇒ UU = UU = 1, as a result of thethe very special conditions (2.26) one has that F ′ = (UF )U−1 = U(FU−1) =UFU−1 = UFU is unambiguously defined.

Dropping the spacetime indices for convenience in the expressions for Fµν , Fµν ,and by repeated use of eqs-(2.25-27), when U−1 = U , the action density is alsoinvariant under finite gauge transformations of the form

< F ′ F ′ > = Re [F ′ F ′] = Re [(UFU−1) (UFU−1)] = Re [(UF ) ( U−1 (UF U−1) )] =

Re [(U F ) (U−1 U) (FU−1)] = Re [(UF ) (FU−1)] = Re [(FU−1) (UF )] =

Re [F ( U−1 (U F ) )] = Re [F (U−1U) F ] = Re [F F ] = Re [F F ] = < F F > .(2.28)

Since the action (2.21) is invariant under finite and infinitesimal ternarygauge transformations, this means that S[Aa

µ; ga] = S[(Aaµ)′; (ga)′ = Ca], where

C = Caea is a constant octonionic-valued coupling which can be obtained fromgauging the octonionic-valued coupling function g(x) to a constant C. This canbe attained by performing a finite gauge transformation with U = U−1 suchthat C = U(x)g(x)U−1(x) ⇒ g(x) = U−1(x)CU(x) and whose components arega =< ea(U−1(x)CU(x)) >. Because the real parts go = Co remain invariantone may identify go = Co with a physical coupling constant. The physicalinterpretation of the remaining 7 vector charges/couplings Ci, i = 1, 2, 3, ....7deserves further investigation.

As is well known, the ordinary 2-bracket does not obey the Jacobi identity

[ ei, [ ej , ek ] ] + [ ej , [ ek, ei ] ] + [ ek, [ ei, ej ] ] = 3 dijkl el 6= 0 (2.29)

If one has the ordinary Yang-Mills expression for the field strength

Fµν = ∂µAν − ∂νAµ + [ Aµ, Aν ] (2.30)

because the 2-bracket does not obey the Jacobi identity, one has an extra (spu-rious) term in the expression for

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[ Dµ, Dν ] Φ = [ Fµν , Φ ] + ( Aµ, Aν , Φ ) (2.31)

given by the crucial contribution of the non-vanishing associator (Aµ, Aν ,Φ) =(AµAν)Φ − Aµ(AνΦ) 6= 0. For this reason, due to the non-vanishing condition(2.29), the ordinary Yang-Mills field strength does not transform homogeneouslyunder ordinary gauge transformations involving the parameters Λ = Λaea

δAµ = ∂µΛ + [Aµ,Λ] (2.32)

and it yields an extra contribution of the form

δFµν = [Fµν ,Λ] + ( Λ, Aµ, Aν) (2.33)

As a result of the additional contribution (Λ, Aµ, Aν) in eq-(2.33), the ordinaryYang-Mills action S =

∫< FµνFµν > will no longer be gauge invariant. Under

infinitesimal variations eqs-(2.33), the variation of the action is no longer zerobut receives spurious contributions of the form δS = −4F l

µνΛiAµjAνkdijkl 6= 0due to the non-associativity of the octonion algebra.

Antisymmetric tensor field theories based on octonionic valued fields Aµν =Aa

µνea, Aµν = −Aνµ can also be constructed. The rank-three octonionic-valuedantisymmetric tensor field strength is defined in terms of the 3-bracket as

Fµνρ = ∂µ Aνρ + ∂ν Aρµ + ∂ρ Aµν +

[ Aµ, Aνρ, g ] + [ Aν , Aρµ, g ] + [ Aρ, Aµν , g ] (2.34)

There is another rank-three tensor given by

Hµνρ = [ Aµ, Aν , Aρ ]. (2.35)

which is only anti-symmetric in the first pair of indices Hµνρ = −Hνµρ. Since the3-bracket obeys the Fundamental identity, under ternary gauge transformations

δAµ = Λab [ ea, eb, Aµ ], δAµν = Λab [ ea, eb, Aµν ] (2.36)

one has that

δFµνρ = Λab [ ea, eb, Fµνρ ], δHµνρ = Λab [ ea, eb, Hµνρ ] (2.37)

if, and only if, the bivector gauge parameters obey the ”self-duality” conditionsΛab = 1

2dabcdΛcd as shown in [28]. An invariant action involving the octonionicvalued fields Fµνρ and Hµνρ in D-dim is of the form

S =1

2κ2

∫dDx <

13!

Fµνρ Fµνρ +12!

Hµνρ Hµνρ > (2.38)

where κ is a parameter with the suitable dimensions to render the action di-mensionless.

8

To finalize this section we discuss further constructions, like having anoctonionic-valued and SU(N)-valued gauge field Aµ = Aam

µ (ea⊗Tm) involvingthe SU(N) algebra generators Tm,m = 1, 2, 3, ...., N2−1 and the octonion alge-bra generators ea, a = 0, 1, 2, 3, ...., 7; i.e. one has octonionic-valued componentsfor the SU(N) gauge fields. The commutator is

[ Aµ, Aν ] = [ Aamµ (ea ⊗ Tm), Abn

ν (eb ⊗ Tn) ] =

12

Aamµ Abn

ν ea, eb ⊗ [Tm, Tn] +12

Aamµ Abn

ν [ea, eb]⊗ Tm, Tn (2.38)

whereea, eb = − 2 δab eo, [ea, eb] = 2 cabc ec (2.39)

and

Tm, Tn =1N

δmn + dmnp Tp, [Tm, Tn] = fmnp Tp (2.40)

One may note that the r.h.s of (2.38) involves both commutators and anti-commutators. Due to the fact that the octonion algebra does not obey theJacobi identities this will spoil the gauge invariance of typical Yang-Mills actionsas described before. Let us have instead a ternary Lie algebra (3-Lie algebra)obeying the ternary commutation relations

[ Tm, Tn, Tp ] = fmnpq Tq (2.41)

and such that the ternary-bracket structure-constants fmnpq obey the fun-damental identity. A 3-Lie-algebra and octonionic-valued field is defined byAµ ≡ Ama

µ (Tm ⊗ ea). However, the triple commutator

[ Aµ, Aν , Aρ ] = [ Amiµ (Tm ⊗ ei), Anj

ν (Tn ⊗ ej), Apkρ (Tp ⊗ ek) ] (2.42)

would furnish a very complicated expression for the r.h.s of eq-(2.42). To sim-plify matters one could define the ternary bracket as

[ Aµ, Aν , Aρ ] = Amiµ Anj

ν Apkρ [Tm, Tn, Tp]⊗ [ei, ej , ek] =

Amiµ Anj

ν Apkρ fmnpq fijkl (Tq ⊗ el) (2.43)

so that one has closure in the r.h.s of (2.43). It is warranted to explore fur-ther these generalized ternary gauge field theories involving 3-Lie algebras andoctonions.

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3 Octonionic Gravity

In this section we shall generalize the octonionic gravity construction basedon the split octonion algebra Os (which strictly speaking is not a division al-gebra) [18], [19] to the full fledged octonion division algebra O. Gµν is anoctonionic-valued metric (Gµν)oeo + (Gµν)iei obeying the Hermiticity condi-tion G†

µν = Gνµ = Gµν , and from which one can infer that the real part ofthe metric is symmetric (G(µν))o, and the 7 imaginary components are anti-symmetric (G[µν])i in their µ, ν indices. The bar denotes octonionic ”complex”conjugation : eo = eo; ei = −ei; i = 1, 2, 3, ....., 7. The diagonal components ofGµν are comprised of real-valued entries; and the off-diagonal ones are com-prised of octonionic-valued entries.

Furthermore, instead of having octonionic-valued metric functions of theform [18] (Gµν)o(xρ) eo + (Gµν)i(xρ) ei, where xρ are ordinary real-valuedcoordinates of a real manifold M whose real-dimension is dimR(M) = D, wehave octonionic-valued metric functions of the form

Gµν = (Gµν)o(Zρ, Zρ) eo + (Gµν)i(Zρ, Zρ) ei (3.1a)

Gµν = (Gµν)o(Zρ, Zρ) eo + (Gµν)i(Zρ, Zρ) ei (3.1b)

Gµν = (Gµν)o(Zρ, Zρ) eo + (Gµν)i(Zρ, Zρ) ei (3.1c)

Gµν = (Gµν)o(Zρ, Zρ) eo + (Gµν)i(Zρ, Zρ) ei (3.1d)

where Zρ, Zρ are octonionic-valued coordinates. The (real and 7 imaginary)components of the octonionic-valued metric

[(Gµν)o(Zρ, Zρ), (Gµν)i(Zρ, Zρ)]; [(Gµν)o(Zρ, Zρ), (Gµν)i(Zρ, Zρ)]; ......(3.2)

are real-valued functions. In the bi-octonions case C×O, one can have complex-valued functions for the components of the metric in eq-(3.2) with i = 1, 2, 3, ...., 7.For the time being we concentrate in the octonions case.

The determinants of non-Hermitian matrices over the division algebras H,Oare not well defined. However the determinant of a 2 × 2 and 3 × 3 Hermitianmatrix over H,O is well defined and real-valued [23]. In the 3 × 3 Hermitianmatrix X case one requires to use the Freudenthal’s determinant definition givenby the trace of the cubic form detX = 1

3Tr(X ∗J (X ×F X)) in terms of the theJordan nonassociative (but commutative) ∗J product and the Freudental ×F

product. This is one of the reasons why it is important to impose the Hermiticitycondition on the octonionic-valued metric Gµν . The indices µ, ν range over thenumber of plausible octonionic dimensions D = 1, 2, 3 where a determinant canbe defined and correspond to 8, 16, 24 real dimensions, respectively.

If one has now an octonionic-valued metric Gµν 6= Gνµ instead of the real-valued metric ηµν = ηνµ, due to the nonassociativity and noncommutativity,

10

the real-valued metric interval ds2 is defined to be

ds2 =12

[ (dZµ Gµν) dZν + dZµ (Gµν dZν) ] +

12

[ (dZν Gµν) dZµ + dZν (Gµν dZµ) ] +

12

[ (dZµ Gµν) dZν + dZµ (Gµν dZν) ] +

12

[ (dZν Gµν) dZµ + dZν (Gµν dZµ) ] (3.3)

where the octonionic (and octonionic-complex conjugate) coordinates are

Zµ = (Zµ)o eo + (Zµ)i ei, µ = 1, 2, 3, ....., D (3.4a)

Zµ = (Zµ)o eo − (Zµ)i ei, µ = 1, 2, 3, ....., D (3.4b)

The components (Zµ)o, (Zµ)i; i = 1, 2, 3, ...., 7 are real-valued entries. In the bi-octonions case C×O they could be complex-valued. The interval ds2 in eq-(3.3)is comprised of sums of terms involving pairs of octonionic-complex conjugatesand for this reason it is real-valued. The first two terms, and the last two termsof (3.3), are respective pairs of octonionic-complex conjugates. For example, inordinary complex manifolds one has a real-valued interval

ds2 = gµν dzµ dzν + gµν dzµ dzν + gµν dzµ dzν + gµν dzµ dzν (3.5)

due to the conditions under complex conjugation (gµν)∗ = gµν , (gµν)∗ = gµν ,(gµν)∗ = gµν , the interval (3.5) is comprised of sums of terms involving pairs ofcomplex conjugates and for this reason it is real-valued.

One may define the analog of a phase rotation or unitary transformation interms of

U = e Λi(Zµ,Zµ) ei , i = 1, 2, 3, ........, 7 (3.6)

where Λi(Zµ, Zµ), for i = 1, 2, 3, ..., 7, are 7 real-valued functions of the oc-tonionic spacetime coordinates Zµ, Zµ;µ = 1, 2, 3, ......, D; i.e. the functionsΛi(Zµ, Zµ) under octonionic conjugation ei = −ei, eo = eo, obey the conditionsΛi(Zµ,Zµ) = Λi(Zµ, Zµ) due to the reality condition imposed on the Λi. As aresult, one has that U satisfies the condition U−1 = U which is the analog of aunitary transformation (unitary matrix). A rigorous definition of an octonionicexponential function, an octonionic Taylor expansion, the Fourier transform andthe Paley-Wiener theorem was provided by [20].

In section 2 ...... we learned from the Moufang identities that when U−1 =U , the new coordinate

Z ′µ = (U Zµ) U = U (Zµ U) = U Zµ U = U Zµ U−1 (3.7)

are unambiguously defined. Dropping the spacetime indices, and the bold facenotation for convenience in the octonionic-valued coordinates Zµ,Zµ , one can

11

again show, after a repeated use of eqs-(2.25-2.27) involving the Moufang iden-tities, that the interval

ds2 = ηµν < dZµ dZν > = < dZµ dZµ > (3.8)

is invariant under the U -transformations (3.7) when U−1 = U ,

(ds′)2 = < dZ ′ dZ ′ > = Re [dZ ′ dZ ′] = Re [(UdZU−1) (UdZU−1)] =

Re [(UdZ) ( U−1 (UdZ U−1) )] = Re [(U dZ) (U−1 U) (dZU−1)] =

Re [(UdZ) (dZU−1)] = Re [(dZU−1) (UdZ)] =

Re [dZ ( U−1 (U dZ) )] = Re [dZ (U−1U) dZ] = Re [dZ dZ] = Re [dZ dZ] =

< dZ dZ > = ds2 (3.9)

Therefore the interval ds2 = (ds′)2 remains invariant under the U -transformations(3.7).

The octonionic-valued connection can be decomposed as

Yσµρ = (Γσ

µρ)o eo + (Θσ

µρ)i ei (3.10)

There are other components Yσµρ, Yσ

µρ, Yσµρ, ..... that must be included as well.

For simplicity we shall not write them down. In complex Hermitian manifoldsone has gµν = gµν = 0, and gµν , gµν are not zero [21]. The only non-vanishingconnection components are Γσ

µρ; Γσµρ; the only non-vanishing curvature compo-

nents are Rσµνρ;R

σµνρ. Lowering indices with the non-vanishing metric com-

ponents gτσ, gτσ yields the non-vanishing curvature components Rτµνρ;Rστµνρ.

The Ricci tensor is Rµν . In the octonionic case matters are more complicateddue to nonassociativity/noncommutativity.

If one restricts the internal part of the octonionic connection in the followingform Θσ

µρ = δσρ Θµ then eq- (3.10) becomes

Yσµρ = (Γσ

µρ)o eo + δσ

ρ (Θµ)i ei (3.11)

The scalar (real) part is given by the spacetime connection

Γσµρ = Γσ

(µρ) + Γσ[µρ] (3.12)

comprised of a symmetric Γσ(µρ) and antisymmetric (torsion) piece T σ

µρ = Γσ[µρ].

The internal (purely imaginary) part of the connection is given by δσρ Θi

µ ei, withi = 1, 2, 3, ....., 7, so that the commutator becomes [Θµ,Θν ] = 2 Θi

µ Θjν cijk ek.

The octonionic-valued curvature, when one restricts the internal part of theconnection to be Θσ

µρ = δσρ Θµ, is given by

Rσµνρ = Rσ

µνρ eo + ( Pσµνρ )k ek ⇒

Rσµνρ = ∂µ Γσ

νρ − ∂ν Γσµρ + Γσ

µτ Γτνρ − Γσ

ντ Γτµρ (3.13)

12

is the standard real part of the curvature. The internal (purely imaginary) partof the curvature tensor can be written in terms of

Pµν = ∂µ Θν − ∂ν Θµ + [ Θµ, Θν ] =

( ∂µ Θkν − ∂ν Θk

µ ) ek + 2 Θiµ Θj

ν cijk ek. (3.14)

such thatPσ

µνρ = ( Pσµνρ )k ek = δσ

ρ (Pµν)k ek =

δσρ ( ∂µ Θν − ∂ν Θµ + [ Θµ, Θν ] )k

ek, ek = e1, e2, e3, ......, e7. (3.15)

By derivatives in eqs-(3.13-3.15) it is understood that ∂µ = (∂/∂Zµ), ∂µ =(∂/∂Zµ). There are other components of the curvature involving derivatives ofthe remaining connection components Yσ

µρ, Yσµρ, ..... that must be included as

well and yielding Rσµνρ,R

σµνρ, ........

When one does not restrict the internal part of the connection to be Θσµρ =

δσρ Θµ = δσ

ρ Θiµei, but instead if it is given by Θσ

µρ = (Θσµρ)

iei, then the expressionfor the curvature is more complicated. In this case, due to cijk = −cjik one hasmodified contributions to the curvature of the form

(Θσµτ )i ei (Θτ

νρ)j ej − (Θσ

ντ )j ej (Θτµρ)

i ei =

−δij

((Θσ

µτ )i (Θτνρ)

j − (Θσντ )j (Θτ

µρ)i

)eo +(

(Θσµτ )i (Θτ

νρ)j + (Θσ

ντ )j (Θτµρ)

i)cijk ek (3.16)

Namely, the real part of the curvature will receive an additional contributionto the prior real part in eq-(3.13) which is given by the second line of eq-(3.16).And the third line of eq-(3.16) differs now from the commutator term δσ

ρ [Θµ,Θν ]in eq-(3.15).

Covariant derivatives are defined by

∇µ Vρ = ∂µ Vρ + Γρσµ Vσ + [ Θµ, Vρ ] (3.17a)

∇µ Vρ = ∂µ Vρ + Γρσµ Vσ + [ Θµ, Vρ ] (3.17b)

etc .... For instance, the commutator will receive additional contributions

[ ∇µ, ∇ν ] Vτ = Rτµνσ Vσ + Γσ

[µν] (∇σ Vτ ) +

[ Pµν , Vτ ] + 6 ( Θµ, Θν , Vτ ) (3.18)

One may notice the contribution of the non-vanishing associator (Θµ,Θν ,Vτ ) tothe r.h.s of eq-( 3.18) . There is also the contribution of the term [Pµν ,Vτ ] due tothe noncommutativity of the octonions, as well as the contribution Γσ

[µν](∇σVτ ).The antisymmetric part of the connection Γσ

[µν] contributes to a non-vanishingtorsion.

In ordinary Riemannian geometry the Bianchi identities ∇[ρRτµν]σ = 0 are

obeyed as well as Rτ[µνσ] = 0 [22], after an antisymmetrization of three indices is

performed. However, due to the non-associativity of octonions, this is no longer

13

the case for the octonionic-valued curvature tensor Rτµνσ . In particular the

non-vanishing associator of three covariant derivatives acting on an octonionic-valued vector is of the form

( ∇µ, ∇ν , ∇ρ ) Vτ = a1 ∇[ρRτµν]σ Vσ + a2 Rσ

[µνρ] ∇σ Vτ 6= 0 (3.19)

the numerical coefficients a1, a2 are real-valued. The antisymmetrization withrespect to the indices [µνρ] is a result of the total antisymmetry of the associatorstructure constant dijkl, for example

( Θµ, Θν , Θρ ) = 2 Θiµ Θj

ν Θkρ dijkl el (3.20)

There are many possible contractions of Rτµνσ,Rτ

µνσ, ....... due to the non-commutativity/nonassociativity of octonions and to the position of the indicesin the metric because it is not symmetric. It obeys the Hermiticity conditionG†

µν = Gνµ = Gµν . Therefore, there are many possible contractions from thefamily of possible octonionic-valued Ricci tensors to obtain octonionic-valuedcurvature scalars. For example, given Rµσ = δν

τ Rτµνσ one has

Rµσ Gσµ 6= Rµσ Gµσ 6= Gσµ Rµσ 6= Gµσ Rµσ (3.21)

As a result of the identities < XY > = < Y X > = Re[XY ] = Re[Y X]one has

< Rµσ Gσµ > = < Gσµ Rµσ > (3.22)

and which is not equal to

< Rµσ Gµσ > = < Gµσ Rµσ > (3.23)

Because there is torsion, and nonmetricity in general ∇ρGµν 6= 0;∇ρGµν 6=0; ......... the Ricci tensor can be decomposed into a symmetric R(µσ) and an-tisymmetric piece R[µσ], for example. If one sets the torsion and nonmetricityto zero, it will yield a relationship among the metric and the connection as inordinary Riemann geometry.

A real-valued analog of the Einstein-Hilbert Lagrangian L = R shouldinvolve sums of all the possible contractions of the Ricci tensors plus theiroctonionic-complex conjugates

L = c1 (Rµσ Gσµ + Gσµ Rµσ) + c2 (Rµσ Gσµ + Gσµ Rµσ) +

c3 (Rµσ Gµσ + Gµσ Rµσ) + c4 (Rµσ Gµσ + Gµσ Rµσ) +

d1 (Rµσ Gσµ + Gσµ Rµσ) + d2 (Rµσ Gµσ + Gµσ Rµσ) +

d3 (Gσµ Rµσ + Rµσ Gσµ) + d4 (Gµσ Rµσ + Rµσ Gµσ) (3.24)

where c1, c2, c3, c4, d1, d2, d3, d4 are real numerical coefficients. From (3.24) onemay notice that in this octonionic case one can construct 8 different real-valued

14

curvature scalars from the contractions of the Ricci curvature tensors due to thenonassociativity and noncommutativity. It is likely that due to symmetries notall of these 8 quantities are independent from each other.

By analogy with what occurs in complex Hermitian manifolds endowed witha Hermitian metric, if the non-vanishing components of the octonionic metricare Gµν ,Gµν one can define the determinant in D = 2, 3 octonionic dimensionsas

det (G) =√

det (Gµν) det (Gµν) (3.25)

and the analog of the Einstein-Hilbert action is

116πGN

∫[dΩ]

√|det(G)| L (3.26)

where the (real-valued) L in this particular case is obtained after setting the lasttwo lines in eq-(3.24) to zero. [dΩ] is a suitable measure in an octonionic D-dimspacetime. The values D = 2, 3 have a correspondence to 16, 24 real-dimensions,respectively. For example, in D = 3 octonionic dimensions, the measure is

(dx0∧dx1∧ ....... ∧dx7) ∧ (dy0∧dy1∧ ....... ∧dy7) ∧ (dz0∧dz1∧ ....... ∧dz7).(3.27)

The components of the metric G = Gµν are

G ≡

g11 go(12)eo + gi

[12]ei go(13)eo + gi

[13]ei

go(21)eo + gi

[21]ei g22 go(23)eo + gi

[23]ei

go(31)eo + gi

[31]ei go(32)eo + gi

[32]ei g33

=

g11 go(12)eo + gi

[12]ei go(13)eo + gi

[13]ei

go(12)eo − gi

[12]ei g22 go(23)eo + gi

[23]ei

go(13)eo − gi

[13]ei go(23)eo − gi

[23]ei g33

(3.28)

It is clear from the second matrix in (3.28) that it is Hermitian from the octo-nionic point of view. The 3× 3 Hermitian matrix G has the same structure as aJordan matrix belonging to the exceptional Jordan-Albert algebra J3[O]. Thereal-valued Freudenthal determinant is given by the trace of the cubic form

det G =13

Tr( G ∗J ( G ×F G ) ) (3.29)

The nonassociative but commutative Jordan product of two Jordan matricesX, Y is

X ∗J Y =12

( X Y + Y X ) (3.30)

15

The (symmetric) Freudenthal product is

X ×F Y = X ∗J Y −12

[ Y Tr(X) + X Tr(Y ) ] +12

[ Tr(X) Tr(Y )− Tr(X∗JY ) ] 1

(3.31)the last term involves the unit matrix 1. Hence, the Freudental determinantallows to construct the Einstein-Hilbert action (3.26) in this case. In D = 2octonionic-dimensions one has the Exceptional Jordan algebra J2[O] involving2× 2 Hermitian matrices with a well defined determinant.

Similar 3×3 Hermitian matrix representations as eq-(3.28) exist for the othercomponents of the octonionic metric as described by eqs-(3.1) and obeying therelations Gµν = Gµν = Gνµ; Gµν = Gµν = Gνµ; Gµν = Gµν = Gνµ. TheirFreudenthal determinant will be computed in the same way as in eq-(3.29).In particular, this will allow us to evaluate the expression for the measure ineq-(3.25) to be used in the action (3.26) for the very special case that the non-vanishing components of the metric are Gµν ,Gµν .

One could add torsion and curvature squared terms to the action (3.26),and other terms if one wishes, like a cosmological constant and derivatives ofcurvature terms. At the moment we shall focus on the simplest of actions linearin the curvature. Clearly, due to the nonassociativity and noncommutativity ofoctonions there are clear generalizations and modifications of ordinary gravity.

Before we finalize this section one ought to mention what is the analog of therotation and Lorentz group when octonions and Exceptional Jordan algebras arethe ingredients of a physical model. There are several ways in which Lie groupscan be defined by Jordan algebras. The analog of the rotation group is theautomorphism group of a Jordan algebra consisting of linear transformationspreserving its multiplication table. These transformations can be expressed interms of the associator (A,B,C) = (AB)C −A(BC) as [25]

M ′ = M + ( A, M, B ) +12!

( A, (A, M, B), B ) + ..... (3.32)

where A,B are traceless elements of the Jordan algebra. The transformation(3.32) is just the ”exponentiation” of the infinitesimal transformation δM =(A,M,B). The analog of the Lorentz group is the reduced structure group withinfinitesimal transformations of the form δM = (A,M,B)+C ∗J M where C isalso a traceless element of the Jordan algebra. The finite transformations underthe reduced structure group obtained by ”exponentiation” are defined as [25]

δfiniteM = δM +12!

δ(δM) +13!

(δ(δ(δM))) + .... ⇒

M ′ −M = ( A, M, B ) + C∗JM +12!

[ ( A, [ ( A, M, B ) + C ∗J M ], B ] +

12!

C ∗J [ ( A, M, B ) + C ∗J M ] + .... (3.33)

The automorphism (rotation) and reduced structure group (Lorentz) of theJ3[O] algebra are F4, E6(−26) respectively. Inspired by the magic Tits square,

16

Gunaydin et al [33] extended the automorphism (rotation) and reduced struc-ture (Lorentz) groups to the so-called conformal and quasi-conformal case lead-ing to non-compact forms of E7, E8, respectively. The authors [23] provided adifferent approach to the Lorentz group. In particular in D = 10, the analog ofOctonionic Mobius (nested) transformations and 3-component Cayley spinorswere constructed.

The key question is : can one extract the Standard Model group (the gaugefields) from the internal part δρ

σ(Θµ)iei of the octonionic gravitational connec-tion ? Instead of having pure octonionic gravity based solely on octonions, onecould have the metric and connection fields taking values in a composition alge-bra of the form C×H×O a la Dixon [7]. The complex numbers are related to aU(1) symmetry associated with the circle S1. The quaternions are related to aSU(2) symmetry associated with the 3-sphere S3. The octonions are related toa SU(3) symmetry associated with the 7-sphere S7. The SU(3) is a subgroup ofthe 14-dim automorphism group of the octonions G2 which leaves invariant theidempotents 1

2 (eo ± ie7). One may include the algebra R of the real numberswhich is associated to S0 and corresponds to the two end-points ±1 of a linesegment. It is more reminiscent of a discrete symmetry like C,P, T .

Therefore, a gravitational theory based on composition algebra of the formC ×H × O a la Dixon [7] would encode the Standard Model group SU(3) ×SU(2) × U(1). Naturally, since the octonionic gravity program described inthis section involves D = 2, 3 octonionic dimensions, which correspond to 16, 24real dimensions, it is amenable to a Kaluza-Klein compactification mechanismto generate the Yang-Mills (GUT, Standard Model) group in lower dimensionsfrom the isometry group of the internal space. A Kaluza-Klein theory withoutextra dimensions involving a metric in curved Clifford space was analyzed by[34].

4 Octonionic Branes, Exceptional Jordan Stringsand Membranes

Given an ordinary string’s world-sheet parametrized by the real-valued coordi-nates σ1, σ2 and embedded in an octonionic-valued target spacetime backgroundZµ, µ = 1, 2, 3, .....D, in the form Zµ(σ1, σ2) = (Zµ)o(σ1, σ2) eo + (Zµ)i(σ1, σ2) ei,allows to formulate the octonionic analog of the Eguchi-Schild string action(area-squared) as

S = − T

4

∫d2σ < [ Zµ, Zν PB ] [ Zµ, Zν PB ] > =

T

4

∫d2σ Real

([ Zν , Zµ PB ] [ Zµ, Zν PB ]

)(4.1)

17

where T is the string’s tension and the Poisson bracket (PB) is defined as usual

Zµ, Zν PB = (∂aZµ) (∂bZν) − (∂bZµ) (∂aZν); ..... (4.2)

One should note the ordering of the indices in eq-(4.2) due to the noncommu-tativity. Another (real-valued) action that can be constructed is

S = − T

8

∫d2σ ( [ Zµ, Zν PB ] [ Zµ, Zν PB ] + o.c.c ) . (4.3)

by adding the octonionic-complex conjugate (o.c.c).Despite the nonassociativity, the real parts of the cubic product obey the

equality

Re [ (ei ej) ek ] = Re [ ei (ej ek) ] (4.4)

however the real parts of the quartic products differ

Re [ (ei ej) (ek el) ] 6= Re [ ei (ej ek) el ] (4.5)

For this reason one must specify the ordering of the quartic products in thespecific form as shown in eqs-(4.1, 4.3) .

The analog of the Polyakov-Howe-Tucker action for an octonionic p-braneaction is

S = −Tp

2

∫dp+1σ

√|h| hab

4[ (∂aZµ Gµν) ∂bZν + ∂aZµ (Gµν ∂bZν) + ......... ] +

(p− 1)2

Tp

∫dp+1σ

√|h| (4.6)

where the ellipsis ....... denote the other contributions from the Gµν ,Gµν , .....components of the metric and the Zµ coordinates. The terms inside the bracketsin eq-(4.6) are comprised of sums of terms involving pairs of octonionic-complexconjugates in order to render the action real-valued, exactly as it was donein eq-(3.3) to obtain a real-valued interval ds2. Tp is the p-brane’s tension ofmass dimension (mass)p+1; hab is the auxiliary p + 1-dim world-volume metriccorresponding to the p-brane. When p = 1, one recovers the string action. Afterthe algebraic elimination of the auxiliary p+1-dim world-volume metric hab viaits equations of motion, one will recover the Nambu-Goto-Dirac action for thep-brane. The (real-valued) induced world-volume metric hab is

hab =14

[(∂aZµ Gµν) ∂bZν + ∂aZµ (Gµν ∂bZν) + ∂bZν (Gµν ∂aZµ) + ...........

](4.7)

after inserting this on-shell value for hab back into the action (4.5) hab = hab,it yields the Nambu-Goto-Dirac p-brane action

S = − Tp

∫dp+1σ

√det |hab| . (4.8)

18

Naturally, when the background metric is real-valued Gµν = gµν there is noambiguity in the ordering in eqs-(4.5).

Other constructions of string and membrane actions based on the nonasso-ciative octonion algebra are possible. For example, a nonassociative formulationof Exceptional Jordan bosonic strings in D = 26 was presented by [24] wherethe string embedding coordinates belong to the 3 × 3 matrix elements of the(traceless) Jordan algebra J3(O) and carry internal charges belonging to rep-resentations of the exceptional F4 algebra. The automorphism group of J3(O)is F4. The traceless condition is required to obtain a 26-dim algebra since theExceptional Jordan-Albert algebra J3[O] is 27-dimensional [26].

A construction of the nonassociative Chern-Simons membrane action fromthe large N limit of an Exceptional Jordan Matrix Model, corresponding tothe direct product of the Exceptional Jordan algebra J3(C × O) with the Liealgebra SU(N), was advanced by [27]. Such Chern-Simons membrane actionhad a rigid global E6 invariance. The E6 Exceptional Jordan Matrix Model wasdeveloped by [31] and the F4 Jordan Matrix Model was studied by [32]. In [27]we proposed also that the generalized spacetime coordinates X may belong toa real Freudenthal algebra defined in Zorn matrix notation as(

a J3[O]J3[O] b

)(4.9)

the two real variables a, b lie along the diagonal The dimension of the algebraFr[O] is 2 + 2 × 27 = 2 + 54 = 2 × 28 = 56. In [27] we argued why thelatter 56 dimensions may permit the complexified formulation of the bosonic28-dimensional version of F theory. The connection between the 28-dim bosonicformulation F theory and quaternionic Jordan algebras of degree four J4[H] hasalso been raised by Smith [26]. The automorphism group of the Freudenthalalgebra Fr[O] is E6 which is a Grand Unification group.

The generalized spacetime coordinates X may belong also to the complexifiedFreudenthal algebra : (

a1 + i a2 J3[C ×O]J3[C ×O] b1 + i b2

)(4.10)

with two complex variables a1 + ia2; b1 + ib2 along the diagonal. The dimensionof the algebra Fr[C × O] is 2 × (2 + 54) = 4 × 28 = 112 which may permita quaternionic formulation of the bosonic 28-dimensional version of F theory.The automorphism group of the complexified Freudenthal algebra Fr[C ×O] isE7.

Gunaydin et al [33] have constructed the conformal (like SO(10, 2), ”twotimes” ) and quasi-conformal (like SO(10 + 2, 4) = SO(12, 4), ”four times”)nonlinear realizations of the Exceptional Lie groups E7(7), E8(8) based on the3-grading and 5-grading decompositions of the noncompact groups E7(7) andE8(8), respectively. The 56 dim representation of E7(7) admits the 3-gradingdecomposition under the E6(6) ×D(dilations) subgroup

56 = 1⊕ (27⊕ 27)⊕ 1. (4.11)

19

The 5-grading decomposition of E8(8) w.r.t the subgroup E7(7) ×D is

248 = 1⊕ 56⊕ (133⊕ 1)⊕ 56⊕ 1 (4.12)

There are no quadratic E7(7) invariants in the 56 representation, nevertheless areal quartic E7(7) invariant I4 can be constructed by means of the Freudenthalternary product [X, Y, Z] → W and a skew-symmetric bilinear form < X,Y >as [33]

I4 =148

< [X, X, X], X > = XijXjkXklXli −14XijXijX

klXkl +

196

εijklmnpqXijXklXmnXpq +196

εijklmnpqXijXklXmnXpq. (4.13)

the symplectic invariant of two 56 representations, like the area element in phasespace

∫dp ∧ dq, is given by

< X,Y > = Xij Yij − Xij Y ij (4.14)

where the fundamental 56 dimensional representation of E7(7) is spanned by theanti-symmetric real tensors ( bi-vectors ) Xij , Xij built from SL(8, R) indices1 ≤ i, j ≤ 8 so that 56 = 28 + 28. An SL(8, R) bi-vector has 28 independentcomponents. The triple product [X, Y, Z] is defined as [33]

[X, Y, Z]ij = − 8 Xik Ykl Zlj − 8 Y ik Xkl Zlj − 8 Y ik Zkl X lj − i ↔ j −

2 Y ij Xkl Zkl − 2 Xij Y kl Zkl − 2 Zij Y kl Xkl +12

εijklmnpq Xkl Ymn Zpq (4.15a)

[X, Y, Z]ij = 8 Xik Y kl Zlj + 8 Yik Xkl Zlj + 8 Yik Zkl Xlj − i ↔ j +

2 Yij Zkl Xkl + 2 Xij Zkl Ykl + 2 Zij Xkl Ykl −12

εijklmnpq Xkl Y mn Zpq (4.15b)

Gunaydin et al [33] have shown that one may exhibit a nonlinear realizationof the algebra E8(8) on a 1 + 56 = 57-dimensional real vector space where thegeneralized spacetime coordinates belong to the algebra Fr[O] ⊕ R and suchthat X = (Xij , Xij , x). The X forms the 56 ⊕ 1 representation of E7(7). Thequartic invariant under the action of the E8(8) group is given by

N4 = I4(Xij , Xij) − x2. (4.16)

The finite displacement in the 57-dim generalized spacetime is defined as

20

δ(X, Y ) = (Xij − Y ij , Xij − Yij , x− y + < X,Y > ) = (Zij , Zij , z) (4.17)

the ”light-cone” in 57-dim which is invariant under E8(8) is defined by the nullcondition

N4 [δ(X, Y )] = I4(Zij , Zij) − z2 = 0. (4.18)

The z generalized coordinate in (4.18) was interpreted as ”entropy” by [33].To sum up, Gunaydin et al [33] concluded that a nonlinear realization of E8(8)

on a space of 57 real dimensions is quasi-conformal in the sense that it leavesinvariant the ”light” cone in (4.18). A further complexification leads to a ”lightcone” in C57 dimensions which is invariant under the complex group E8(C).

Okubo, de Wit-Nicolai and Gursey-Tze constructed (independently) an oc-tonionic triple-product among three octonionic x, y, z variables [4]

[ x, y, z ]Okubo =12

( (x, y, z) + < x|eo > [y, z] + < y|eo > [z, x] ) +

12

( < z|eo > [x, y] − < z|[x, y] > eo ) . (4.19)

where eo is the Octonion unit element; < x|y >=< y|x >= Re[xy] = Re[yx] isa symmetric bilinear non-degenerate form, and (x, y, z) = (xy)z − x(yz) is thenon-vanishing associator for the octonionic variables. The triple product (4.19)is totally antisymmetric in x, y, z whereas the triple product (4.15a, 4.15b) isnot. A quartic invariant among four octonionic variables can be defined fromthe triple product (4.19) and the bilinear form as < w|[x, y, z] >. It is totallyantisymmetric in the four variables x, y, z, w. This latter total antisymmetryproperty will permit us to construct a Nambu-Goto-Dirac 3-brane action corre-sponding to a 3+1-dim world volume (our world ?) which is embedded into anoctonionic spacetime background with a real-valued flat metric ηµν , and whoseoctonionic coordinates are Zµ, µ = 1, 2, ....., D. Hence, we propose the following3-brane action on a octonionic flat background

S = −T3

∫d4σ

√[ < ∂σ1Zµ1 | [ ∂σ2Zµ2 , ∂σ3Zµ3 , ∂σ4Zµ4 ] > ]2 (4.20)

T3 is the 3-brane tension. The square [....]2 is

< ∂σ1Zµ1 | [ ∂σ2Zµ2 , ∂σ3Zµ3 , ∂σ4Zµ4 ] > < ∂σ1Zµ1 | [ ∂σ2

Zµ2 , ∂σ3Zµ3 , ∂σ4

Zµ4 ] >(4.21)

where µ1, µ2, µ3, µ4 = 1, 2, 3, ....., D ≥ 4. The quantity inside the square root(4.20) plays the same role in ordinary p-branes actions as the determinant ofthe induced world-volume metric resulting from the embedding of the p+1-dimworld-volume into a flat target spacetime background

21

det hab = det ( ∂aXµ ∂bXν ηµν) =

Xµ1 , Xµ2 , ........., Xµp+1 NPB Xµ1 , Xµ2 , ........., Xµp+1 NPB (4.22)

where the brackets NPB are given by the standard Nambu-Poisson brackets. Ifthe target background is not flat, one must contract the spacetime indices of(4.22) in the following form

Xµ1 , Xµ2 , ........., Xµp+1 Xν1 , Xν2 , ........., Xνp+1 gµ1ν1(X) gµ2ν2(X) ...... gµp+1νp+1(X)(4.23)

An octonionic background endowed with an octonionic-valued metric Gµν willcomplicate the expression inside the square root of eq-(4.20) resulting from thenonassociativity and noncommutativity. For this reason we opted to choose areal-valued flat metric ηµν which permits us to contract spacetime indices in asimple fashion leading to a suitable sum of brackets squared (4.21).

An interesting question would be if the 3-brane action (4.20) involving aquartic product displays any symmetry under E7 when the number of back-ground octonionic dimensions is 7 and corresponding to 8×7 = 56 real-dimensions.The latter 56 real dimensions were required to construct the E7(7) quartic I4

invariant < [X, X, X], X > described by eq-(4.13). For a 3-brane action to dis-play a symmetry under E8 one would require (at least) a background whoseoctonionic dimensions is 8 and corresponding to 8 × 8 = 64 real-dimensions;i.e. the rank 7, 8 algebras E7, E8 would require a background of 7, 8 octonionicdimensions, respectively where the 3-brane lives.

To summarize the main results of this section, we have provided generalizedoctonionic string and brane actions given by eqs-(4.1, 4.3, 4.6, 4.20) that arenovel to our knowledge and raised the possibility that the 3-brane action (4.20)(based on the quartic product) in octonionic flat backgrounds of 7, 8 octonionicdimensions may display a E7, E8 symmetry. We conclude with some final re-marks pertaining to the developments related to Jordan exceptional algebras,octonions, black-holes in string theory and quantum information theory.

The E7 Cartan quartic invariant was used by [36] to construct the entangle-ment measure associated with the tripartite entanglement of seven quantum-bitsrepresented by the group SL(2, C)3 and realized in terms of 2× 2× 2 cubic ma-trices. It was shown by [37] that this tripartite entanglement of seven quantum-bits is entirely decoded into the discrete geometry of the octonion Cayley-Fanoplane. The analogy between quantum information theory and supersymmetricblack holes in 4D string theory compactifications was extended further by [37].The role of Jordan algebras associated with the homogeneous symmetric spacespresent in the study of extended supergravities, BPS black holes, quantum at-tractor flows and automorphic forms can be found in [35]. An extensive reviewof the established relationships between black hole entropy in string theory andthe quantum entanglement of qubits and qutrits in quantum information theorycan be found in [36].

The classification of symmetric spaces associated with the scalars of N ex-tended Supergravity theories, emerging from compactifications of 11D super-gravity to lower dimensions, and the construction of the U -duality groups as

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spectrum-generating symmetries for four-dimensional BPS black-holes [35], [38]also involved exceptional symmetries associated with the exceptional magic Jor-dan algebras J3[R,C,H, O]. The discovery of the anomaly free 10-dim heteroticstring for the algebra E8 × E8 was another hallmark of the importance of Ex-ceptional Lie groups in Physics. Exceptional Jordan strings [24] carry internalcharges that might bear a relation to the charges associated with the BPS statesin the black hole solutions of N = 8 Supergravity. Finally, we may ask if thefour-dim measure (4.20) associated with the 3 brane action, and defined by aninvariant quartic product over the octonions, bears any connection to an entan-glement Entropy function given by the square root of a hyper-determinant [36].All this deserves further investigation.

AcknowledgmentsWe thank M. Bowers for her assistance.

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